Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces
The essential aim of the present paper is to solve a system of generalized mixed variational inequalities without any monotonicity assumption for the mappings in Banach spaces. Formulation: Find \(x^*, y^* \in K\) such that there exist \(u^* \in T_1(y^*)\) and \(v^* \in T_2(x^*)\) satisfying \[ \begin{aligned} \langle u^* + Jx^* - Jy^*, y-x^* \rangle + f_1(y)-f_1 (x^*) \geq 0, \forall y \in K\\ \langle v^* + Jy^* - Jx^*, y-y^* \rangle + f_2(y)-f_2 (y^*) \geq 0, \forall y \in K \end{aligned}\tag{1} \] It is also pointed out, that: if \(f_i (x)=0\) \((i=1,2)\) for all \(x \in K\) (a nonempty closed convex of Banach space \(X\)), the problem (1) is equivalent to the system of generalized variational inequalities: Find \(x^*, y^* \in K\) such that there exist \(u^* \in T_1(y^*)\) and \(v^* \in T_2(x^*)\) satisfying \[ \begin{aligned} \langle u^* + Jx^* - Jy^*, y-x^* \rangle \geq 0, \forall y \in K\\ \langle v^* + Jy^* - Jx^*, y-y^* \rangle \geq 0, \forall y \in K\\ \end{aligned}\tag{2} \] If for any \(\rho, \nu > 0\), \(X=H\) is a Hilbert space, \(T_1y^*=\{\rho g_1 (y^*)\}\) and \(T_2x^*=\{\rho g_2 (x^*)\}\) for all \(x^*, y^* \in K\), where \(g_1\) and \(g_2: K \rightarrow X^*\) are single-valued mappings, then problem (1) reduces to find \(x^*, y^*\) such that \[ \begin{aligned} \langle \rho g_1(y^*)+ x^* - y^*, y-x^* \rangle + f_1(y)-f_1 (x^*) \geq 0, \forall y \in K\\ \langle \eta g_2(x^*)+ y^* - x^*, y-y^* \rangle + f_2(y)-f_2 (y^*) \geq 0, \forall y \in K\end{aligned}\tag{3} \] Main result: If the upper semi-continuous set values mappings \(T_1, T_2: K \rightarrow 2^{x^*}\) with closed values, and the proper convex lower semi-continuous mapping \(f_i: K \rightarrow\mathbb R \cup \{+ \infty \}\) satisfy the assumptions: \(f_i(x) \geq0\) for all \(x \in K\) and \(f_i(0)=0\), for any \(x \in K\) and any \(z_i \in T_i(x)\) (\(i=1,2\)), \(\langle z_i, J^* (Jx-z_i) \rangle \geq0\), the set-valued mappings \(J-T_1\) and \(J-T_2\) are compact and \( \sum^\infty_{n=0} \alpha_n (1- \alpha_n) = \infty\) (\(\alpha_n \in [0,1]\) for all \(n \geq0\)), then the system (1) has a solution \((x^*, y^*)\) and the sequence \(\{x_n\}\) and \(\{y_n\}\) defined by the proposed algorithm (for arbitrary chosen initial points \(x_0, y_0 \in K\) compute the sequence \(\{x_n\}\) and \(\{y_n\}\) by: \(u_n \in T_1y_n\), \(v_n \in T_2x_n\), \(x_{n+1}=(1-\alpha_n) x_n + \alpha_n \pi_K^{fi} (Jy_n - u_n)\), \(y_{n+1} = (Jx_{n+1} - v_{n+1})\), \(\pi_K^f: X^* \rightarrow 2^K\), the generalized \(f\)-projection operator) have convergent subsequences \(\{x_{n_i}\}\) and \(\{y_{n_i}\}\) such that \(x_{n_i} \rightarrow x^*\), \(y_{n_i} \rightarrow y^*\), as \(i \rightarrow +\infty\), respectively. The convergence analysis for the systems (1) and (3) respectively is given as well.
System of generalized mixed variational inequalities
scientific article; zbMATH DE number 5907619
scientific article